∫ −7200x+e2x(125000+30000x+2400x2+64x3)_______________________________

Optimal. Leaf size=26 \[ 4 e^{2 x}-\frac {16 x^2}{\left (\frac {25-x}{3}+x\right )^2} \]

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Rubi [A] time = 0.20, antiderivative size = 20, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {6688, 12, 6742, 2194, 37} \begin {gather*} 4 e^{2 x}-\frac {144 x^2}{(2 x+25)^2} \end {gather*}

Int[(-7200*x + E^(2*x)*(125000 + 30000*x + 2400*x^2 + 64*x^3))/(15625 + 3750*x + 300*x^2 + 8*x^3),x]

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +

1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (-900 x+e^{2 x} (25+2 x)^3\right )}{(25+2 x)^3} \, dx\\ &=8 \int \frac {-900 x+e^{2 x} (25+2 x)^3}{(25+2 x)^3} \, dx\\ &=8 \int \left (e^{2 x}-\frac {900 x}{(25+2 x)^3}\right ) \, dx\\ &=8 \int e^{2 x} \, dx-7200 \int \frac {x}{(25+2 x)^3} \, dx\\ &=4 e^{2 x}-\frac {144 x^2}{(25+2 x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.04, size = 24, normalized size = 0.92 \begin {gather*} 8 \left (\frac {e^{2 x}}{2}-\frac {18 x^2}{(25+2 x)^2}\right ) \end {gather*}

Integrate[(-7200*x + E^(2*x)*(125000 + 30000*x + 2400*x^2 + 64*x^3))/(15625 + 3750*x + 300*x^2 + 8*x^3),x]

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fricas [A] time = 0.76, size = 34, normalized size = 1.31 \begin {gather*} \frac {4 \, {\left ({\left (4 \, x^{2} + 100 \, x + 625\right )} e^{\left (2 \, x\right )} + 900 \, x + 5625\right )}}{4 \, x^{2} + 100 \, x + 625} \end {gather*}

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="fricas

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giac [B] time = 0.21, size = 41, normalized size = 1.58 \begin {gather*} \frac {4 \, {\left (4 \, x^{2} e^{\left (2 \, x\right )} + 100 \, x e^{\left (2 \, x\right )} + 900 \, x + 625 \, e^{\left (2 \, x\right )} + 5625\right )}}{4 \, x^{2} + 100 \, x + 625} \end {gather*}

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="giac")

4*(4*x^2*e^(2*x) + 100*x*e^(2*x) + 900*x + 625*e^(2*x) + 5625)/(4*x^2 + 100*x + 625)

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method	result	size

risch	\(\frac {900 x +5625}{x^{2}+25 x +\frac {625}{4}}+4 \,{\mathrm e}^{2 x}\)	\(24\)
default	\(-\frac {22500}{\left (2 x +25\right )^{2}}+\frac {1800}{2 x +25}+4 \,{\mathrm e}^{2 x}\)	\(26\)
norman	\(\frac {3600 x +2500 \,{\mathrm e}^{2 x}+400 x \,{\mathrm e}^{2 x}+16 \,{\mathrm e}^{2 x} x^{2}+22500}{\left (2 x +25\right )^{2}}\)	\(36\)

int(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x,method=_RETURNVERBOSE)

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {8 \, {\left (4 \, x^{3} + 150 \, x^{2} + 1875 \, x\right )} e^{\left (2 \, x\right )}}{8 \, x^{3} + 300 \, x^{2} + 3750 \, x + 15625} + \frac {900 \, {\left (4 \, x + 25\right )}}{4 \, x^{2} + 100 \, x + 625} - \frac {62500 \, e^{\left (-25\right )} E_{3}\left (-2 \, x - 25\right )}{{\left (2 \, x + 25\right )}^{2}} - 375000 \, \int \frac {e^{\left (2 \, x\right )}}{16 \, x^{4} + 800 \, x^{3} + 15000 \, x^{2} + 125000 \, x + 390625}\,{d x} \end {gather*}

integrate(((64*x^3+2400*x^2+30000*x+125000)*exp(x)^2-7200*x)/(8*x^3+300*x^2+3750*x+15625),x, algorithm="maxima

8*(4*x^3 + 150*x^2 + 1875*x)*e^(2*x)/(8*x^3 + 300*x^2 + 3750*x + 15625) + 900*(4*x + 25)/(4*x^2 + 100*x + 625)

- 62500*e^(-25)*exp_integral_e(3, -2*x - 25)/(2*x + 25)^2 - 375000*integrate(e^(2*x)/(16*x^4 + 800*x^3 + 1500

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mupad [B] time = 0.10, size = 20, normalized size = 0.77 \begin {gather*} 4\,{\mathrm {e}}^{2\,x}+\frac {3600\,x+22500}{{\left (2\,x+25\right )}^2} \end {gather*}

int(-(7200*x - exp(2*x)*(30000*x + 2400*x^2 + 64*x^3 + 125000))/(3750*x + 300*x^2 + 8*x^3 + 15625),x)

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sympy [A] time = 0.13, size = 22, normalized size = 0.85 \begin {gather*} - \frac {7200 \left (- 4 x - 25\right )}{32 x^{2} + 800 x + 5000} + 4 e^{2 x} \end {gather*}

integrate(((64*x**3+2400*x**2+30000*x+125000)*exp(x)**2-7200*x)/(8*x**3+300*x**2+3750*x+15625),x)

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3.70.33 \(\int \frac {-7200 x+e^{2 x} (125000+30000 x+2400 x^2+64 x^3)}{15625+3750 x+300 x^2+8 x^3} \, dx\)